Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,6)(3,7,8,5)(10,13)(14,15)(17,18)(19,23)(20,22)(21,24);(1,2,4,6)(3,5,8,7)(9,10,12,13)(11,14,16,15)(17,18,20,22)(19,21,24,23);(1,2)(3,5)(4,6)(7,8)(17,18)(19,21)(20,22)(23,24);(3,8)(5,7)(9,10,12,13)(11,15,16,14)(17,19,20,24)(18,21,22,23);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72232285, #72232833, #72300627, … |
| maximal normal subgroup of | #72232285, #72232833, #72300627, … |
| maximal quotient of | #72232380, #72232956, #72300665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315261);